Simulation Based Inference in Simultaneous Equations

In the context of multivariate regression (MLR) and simultaneous equations (SE), it is well known that commonly employed asymptotic test criteria are seriously biased towards over-rejection. In this paper, we propose exact likelihood based tests for possibly nonlinear hypotheses on the coefficients of SE systems. We discuss a number of bounds tests and Monte Carlo simulation based tests. The latter involves maximizing a randomized p-value function over the relevant nuisance parameter space which is done numerically by using a simulated annealing algorithm. We consider limited and full information models, in which case we introduce a multi-equation Anderson-Rubin-type test. Illustrative Monte Carlo experiments show that: (i) bootstrapping standard instrumental variable (IV) based criteria fails to achieve size control, especially (but not exclusively) under near non-identification conditions, and (ii) the tests based on IV estimates do not appear to be boundedly pivotal and so no size-correction may be feasible. By contrast, likelihood ratio based tests work well in the experiments performed.

Estimating New Keynesian Phillips Curves Using Exact Methods

The authors use simple new finite-sample methods to test the empirical relevance of the New Keynesian Phillips curve (NKPC) equation. Unlike tests based on the generalized method of moments, the generalized Anderson-Rubin (1949) tests are immune to the presence of weak instruments and allow, by construction, the identification status of a model to be assessed. The authors illustrate their results using Gali and Gertler's (1999) NKPC specifications and data, as well as a survey-based inflation-expectation series from the Federal Reserve Bank of Philadelphia. The test the authors use rejects Gali and Gertler's estimates (conditional on the latters' choice of instruments). Nevertheless, and in contrast with results obtained by Ma (2002), the authors do obtain relatively informative confidence sets. This provides support for NKPC equations and illustrates the usefulness of using exact procedures in estimations based on instrumental variables. The authors' results also reveal that the least well-identified parameter is w; namely, the proportion of firms that do not adjust their prices in period t.Econometric and statistical methods; Inflation and prices

Structural Change in Covariance and Exchange Rate Pass-Through: The Case of Canada

The authors address empirically the implications of structural breaks in the variance-covariance matrix of inflation and import prices for changes in pass-through. They define pass-through within a correlated vector autoregression (VAR) framework as the response of domestic inflation to an impulse in import price inflation. This approach allows them to examine changes in both the amount and the duration of pass-through.Econometric and statistical methods

Testing the Stability of the Canadian Phillips Curve Using Exact Methods

Postulating two different specifications for the Canadian Phillips curve (a purely backwardlooking model, and a partly backward-, partly forward-looking model), the authors test for structural breaks in the parameters of the equation. In each case, they account for the possibilities that: (i) breaks can be discrete, or continuous, and (ii) available data samples may be too small to justify using asymptotically valid structural-change tests. Thus, the authors use recent testing procedures that are valid in finite samples, applying the Dufour-Kiviet (1996) methodology for discrete-type breaks, and the Dufour (2002) Maximized Monte Carlo test method for continuous-type shifts. The second test accounts for nuisance parameters that appear only under the alternative. The proposed alternative is a Kalman-filter-based time-varying-parameter specification, with coefficients that follow random walks. The authors find evidence for linear and non-linear breaks, the latter being characterized by continuous and unpredictable-type shifts in the inflation-dynamics coefficients.Econometric and statistical methods

Finite-Sample Simulation-Based Tests in Seemingly Unrelated Regressions

In this paper, we propose finite and large sample likelihood based test procedures for possibly non-linear hypotheses on the coefficients of SURE systems. Two complementary approaches are described. First, we propose an exact Monte Carlo bounds test based on the standard likelihood ratio criterion. Second, we consider alternative Monte Carlo tests which can be run whenever the bounds are not conclusive. These include, in particular, quasi-likelihood ratio criteria based on non-maximum-likelihood estimators. Illustrative Monte Carlo experiments show that: (i) the bounds are sufficiently tight to yield conclusive results in a large proportion of cases, and (ii) the randomized procedures correct all the usual size distortions in such contexts. The procedures proposed are finally applied to test restrictions on a factor demand model.Multivariate Linear Regression, Seemingly Unrelated Regressions, Monte Carlo Test, Bounds Tests, Nonlinear Hypothesis, Finite-Sample Test, Exact Test, Bootstrap, Factor Demand, Cost Function

Exact Tests for Contemporaneous Correlation of Disturbances in Seemingly Unrelated Regressions

This paper proposes finite-sample procedures for testing the SURE specification in multi-equation regression models, i.e. whether the disturbances in different equations are contemporaneously uncorrelated or not. We apply the technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] to obtain exact tests based on standard LR and LM zero correlation tests. We also suggest a MC quasi-LR (QLR) test based on feasible generalized least squares (FGLS). We show that the latter statistics are pivotal under the null, which provides the justification for applying MC tests. Furthermore, we extend the exact independence test proposed by Harvey and Phillips (1982) to the multi-equation framework. Specifically, we introduce several induced tests based on a set of simultaneous Harvey/Phillips-type tests and suggest a simulation-based solution to the associated combination problem. The properties of the proposed tests are studied in a Monte Carlo experiment which shows that standard asymptotic tests exhibit important size distortions, while MC tests achieve complete size control and display good power. Moreover, MC-QLR tests performed best in terms of power, a result of interest from the point of view of simulation-based tests. The power of the MC induced tests improves appreciably in comparison to standard Bonferroni tests and in certain cases outperform the likelihood-based MC tests. The tests are applied to data used by Fischer (1993) to analyze the macroeconomic determinants of growth. Cet article propose des procédures exactes pour tester la spécification SURE (régressions empilées) dans le contexte des régressions linéaires multivariées, i.e. si les perturbations des différentes équations sont corrélées ou non. Nous appliquons la technique des tests de Monte Carlo (MC) [Dwass (1957), Barnard (1963)] pour obtenir des tests d'indépendance exacts fondés sur les critères du quotient de vraisemblance (LR) et du multiplicateur de Lagrange (LM). Nous suggérons aussi un critère du type quasi-quotient de vraisemblance (QLR) dérivé sur base des moindres carrés généralisés réalisables (FGLS). Nous démontrons que ces statistiques sont libres de paramètres de nuisance sous l'hypothèse nulle, ce qui justifie l'application des tests de Monte Carlo. Par ailleurs, nous généralisons le test exact proposé par Harvey et Phillips (1982) au contexte des équations multiples. En particulier, nous proposons plusieurs tests induits basés sur des tests de type Harvey-Phillips et nous suggérons une technique basée sur des simulations afin de résoudre le problème de combinaison de tests. Nous évaluons les propriétés des tests que nous proposons dans le cadre d'une étude de Monte Carlo. Nos résultats montrent que les tests asymptotiques usuels présentent de sérieuses distorsions de niveau, alors que les tests de MC contrôlent parfaitement le niveau et ont une bonne puissance. De plus, les tests QLR se comportent bien du point de vue de la puissance; ce résultat est intéressant vu que les tests (multivariés) que nous proposons sont basés sur des simulations. La puissance des tests de MC induits augmente sensiblement par rapport aux tests fondés sur l'inégalité de Bonferroni et, dans certains cas, dépasse la puissance des tests de MC fondés sur la vraisemblance. Nous appliquons les tests sur des données utilisées par Fischer (1993) pour analyser des modèles de croissance.Seemingly unrelated regressions, SURE system, multivariate linear regression, contemporaneous correlation, exact test, finite-sample test, Monte Carlo test, bootstrap, induced test, LM test, likelihood ratio test, specification test, macroeconomics, growth, Régressions empilées, système SURE, test d'indépendance, régression linéaire multivariée, corrélation contemporaine, test exact, test à distance finie, test de Monte Carlo, bootstrap, test induit, test LM, quotient de vraisemblance, test de spécification, macroéconomie, croissance

Simulation-Based Finite-Sample Inference in Simultaneous Equations

In simultaneous equation (SE) contexts, nuisance parameter, weak instruments and identification problems severely complicate exact and asymptotic tests (except for very specific hypotheses). In this paper, we propose exact likelihood based tests for possibly nonlinear hypotheses on the coefficients of SE systems. We discuss a number of bounds tests and Monte Carlo simulation based tests. The latter involves maximizing a randomized p-value function over the relevant nuisance parameter space which is done numerically by using a simulated annealing algorithm. We consider limited and full information models. We extend, to non-Gaussian contexts, the bound given in Dufour (Econometrica, 1997) on the null distribution of the LR criterion, associated with possibly non-linear- hypotheses on the coefficients of one Gaussian structural equation. We also propose a tighter bound which will hold: (i) for the limited information (LI) Gaussian hypothesis considered in Dufour (1997) and for more general, possibly cross-equation restrictions in a non-Gaussian multi-equation SE system. For the specific hypothesis which sets the value of the full vector of endogenous variables coefficients in a limited information framework, we extend the Anderson-Rubin test to the non-Gaussian framework. We also show that Wang and Zivot's (Econometrica, 1998) asymptotic bounds-test may be seen as an asymptotic version of the bound we propose here. In addition, we introduce a multi-equation Anderson-Rubin-type test. Illustrative Monte Carlo experiments show that: (i) bootstrapping standard instrumental variable (IV) based criteria fails to achieve size control, especially (but not exclusively) under near non-identification conditions, and (ii) the tests based on IV estimates do not appear to be boundedly pivotal and so no size-correction may be feasible. By contrast, likelihood ratio based tests work well in the experiments performedSimultaneous Equation, Weak Instruments, Monte Carlo test, Identification

Finite-Sample Simulation-Based Tests in Seemingly Unrelated Regressions

In this paper, we propose finite and large sample likelihood based test procedures for possibly non-linear hypotheses on the coefficients of SURE systems. Two complementary approaches are described. First, we propose an exact Monte Carlo bounds test based on the standard likelihood ratio criterion. Second, we consider alternative Monte Carlo tests which can be run whenever the bounds are not conclusive. These include, in particular, quasi-likelihood ratio criteria based on non-maximum-likelihood estimators. Illustrative Monte Carlo experiments show that: (i) the bounds are sufficiently tight to yield conclusive results in a large proportion of cases, and (ii) the randomized procedures correct all the usual size distortions in such contexts. The procedures proposed are finally applied to test restrictions on a factor demand model.Multivariate linear regression, Seemingly unrelated regressions, Monte Carlo test, Bounds test, Nonlinear hypothesis, Finite-sample test, Exact test, Bootstrap, Factor demand, Cost function

Identification Robust Confidence Sets Methods for Inference on Parameter Ratios and their Application to Estimating Value-of-Time

The problem of constructing confidence set estimates for parameter ratios arises in a variety of econometrics contexts; these include value-of-time estimation in transportation research and inference on elasticities given several model specifications. Even when the model under consideration is identifiable, parameter ratios involve a possibly discontinuous parameter transformation that becomes ill-behaved as the denominator parameter approaches zero. More precisely, the parameter ratio is not identified over the whole parameter space: it is locally almost unidentified or (equivalently) weakly identified over a subset of the parameter space. It is well known that such situations can strongly affect the distributions of estimators and test statistics, leading to the failure of standard asymptotic approximations, as shown by Dufour. Here, we provide explicit solutions for projection-based simultaneous confidence sets for ratios of parameters when the joint confidence set is obtained through a generalized Fieller approach. A simulation study for a ratio of slope parameters in a simple binary probit model shows that the coverage rate of the Fieller's confidence interval is immune to weak identification whereas the confidence interval based on the delta-method performs poorly, even when the sample size is large. The procedures are examined in illustrative empirical models, with a focus on choice modelsconfidence interval; generalized Fieller's theorem; delta-method; weak identification; ratio of parameters.

Structural Change and Forecasting Long-Run Energy Prices

The authors test the statistical significance of Pindyck’s (1999) suggested class of econometric equations that model the behaviour of long-run real energy prices. The models postulate meanreverting prices with continuous and random changes in their level and trend, and are estimated using Kalman filtering. In such contexts, test statistics are typically non-standard and depend on nuisance parameters. The authors use simulation-based procedures to address this issue; namely, a standard Monte Carlo test and a maximized Monte Carlo test. They find statistically significant instabilities for coal and natural gas prices, but not for crude oil prices. Out-of-sample forecasts are calculated to differentiate between significant models.Econometric and statistical methods